Power–law properties of Chinese stock market

ARTICLE IN PRESS

Physica A 353 (2005) 425–432 www.elsevier.com/locate/physa

Power–law properties of Chinese stock market C. Yana, J.W. Zhanga,b,c,, Y. Zhanga, Y.N. Tanga a

School of Physics, Peking University, Beijing 100871, China Key Laboratory of Quantum Information and Quantum Measurements of Ministry of Education, Beijing 100871, China c Department of Physics, Teachers College, Shihezi University, Xinjiang 832003, China

b

Received 25 September 2004; received in revised form 15 January 2005 Available online 19 March 2005

Abstract Price changes of primary returns of Chinese stock market are analyzed over a period of about 8 years. The probability distribution of relative changes in returns satisfies the power–law form. However, the distribution is not consistent with the analysis of US and other stock markets that seem to contain the exponent of an inverse cube. Furthermore, we find that the positive and negative returns do not behave consistently, which indicates a significant asymmetry in the distribution. r 2005 Elsevier B.V. All rights reserved. PACS: 64.60.L; 89.90.+n; 87.10.+e; 89.65.Gh Keywords: Stock market; Power–law; Asymmetry

1. Introduction Using the methods developed for physical systems, the analysis of financial data has a long tradition [1–4]. In recent years, physicists have focused their interest on using the analysis methods of dynamic, complex systems in modeling the financial Corresponding author.

E-mail address: [email protected] (J.W. Zhang). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.02.010

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and economic processes [5–10]. Recent studies have revealed some interesting findings of the distribution of stock price fluctuations. Results show that it follows a power–law decay with exponents consistent with an inverse cube at the tails (the exponent a ’ 3), which lies outside the Le´vy stable range (0oao2) [11–16]. This rule is regarded as a universal one since stocks in the United States [12,14,17], Germany [11] and Australia [18] all obey it. In addition, some market indices also have this characteristic, such as the index of S&P 500, Dow Jones, NIKKEI, Hang Seng, Milan, and DAX [11,13,15]. Several theoretical models have been proposed [19–21] to explain the mechanism of the empirical power–law distribution. For instance, Solomon and Richmond [19] build a multiagent system by the use of a generalized Lotka–Volterra model; Gabaix et al. [20] present another theory based on the economic optimization by heterogeneous agents. However, further study of the Indian stock market [22] demonstrates another scenario. In the Indian stock market the distribution of daily returns of the stock price PðgÞ decays as an exponential function PðgÞ  expðbgÞ; where g is the normalized return and b is the decay parameter. In addition, Huang [23] found that by skipping the data in the first 20 min of each morning session, the 1-min data of the Hang Seng index show an exponential-type decay as PðxÞ  expðajxjÞ=jxj; where xðtÞ ¼ indexðtÞ  indexðt  DtÞ: These distinct findings complicate the research work of the stock market behavior and diversifies the possible theoretical explanations. In order to test the ubiquity of the inverse-cubic law in the stock markets, we investigate the distribution of daily returns in Chinese stock market. The database of closing prices from the Shanghai Stock Exchange and Shenzhen Stock Exchange is accumulated for calculation [25]. It is observed that the distribution behavior of daily returns follow the power–law rule but the positive and the negative tails show both inconsistence of the exponent of an inverse cube and an asymmetric characteristic.

2. Distribution analysis We analyze the closing price of individual stocks from the Shanghai Stock Exchange and Shenzhen Stock Exchange [24]. In order to cover as many data as possible, we select all the stocks listed before 1 November 1993 in both stock exchanges, and choose the period for analyzing from 3 January 1994 to 31 December 2001. We have excluded data of the first couple of years when both stock exchanges were founded. The total number of selected stocks reaches 104, including 76 from the Shanghai Stock Exchange and 28 from Shenzhen Stock Exchange. In this case the total data reach the scale of around 2  105 : For a time series SðtÞ of prices of a company [25], the return GðtÞ ¼ GDt ðtÞ over a time scale Dt is defined as the forward change in the logarithm of SðtÞ G Dt ðtÞ ¼ ln Sðt þ DtÞ  ln SðtÞ .

(1)

We investigate the time series of returns on time scale Dt ¼ 1 day. In order to make a synthetic analysis of the data from different companies, we use a normalized

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return g as g

G  hGiT , V

(2)

where V ¼ V ðDtÞ is defined through V 2 ¼ hG 2 iT  hGi2T ; and h iT denotes an average over the entire length of the time series. After getting the normalized increments gðtÞ; we first use the power–law fit method to estimate the distribution characteristic. A power–law asymptotic behavior is found which reads Pðg4xÞ  xa .

(3)

Although former studies reveal an inverse-cubic exponent in both price and index distributions [11–15,17,18], our regression fits yield ( 2:44  0:02 ðpositive tailÞ a¼ . (4) 4:29  0:04 ðnegative tailÞ Figs. 1(a) and (b) depict the cumulative distribution of daily returns for the positive and negative tail, respectively. Remarkably the value of a for the positive tail is quite smaller than that for the negative tail, which indicates the asymmetry the positive and negative tails. Besides above results, it is important to note that in both figures at the region around g ’ 3 the continuity of the smoothness and the differential coefficient of the curve are not perfect. According to these figures, the positive and negative tails convert their directions to some extent into a greater and lesser value of the exponent, respectively. This may be caused by the unexpected counterfeit behavior in the market, which usually affects the market more directly in an under-developed economic environment than in a developed one. Another important issue that should be pointed out is the behavior near the end of the tails. At the region gX10; the positive tail shows a trend to follow an asymptotic behavior

10 -1 10

-2

10

-3

positive tail

10 -4 10 -5 10 -6 -1 10

(a)

10 0 P (cumulated probability)

P (cumulated probability)

10 0

0

1

10 10 g (normalized return)

10

10 -1 10

10 -3 10 -4 10 -5 10 -6 -1 10

2

(b)

negative tail

-2

0

1

10 10 g (normalized return)

10

2

Fig. 1. Cumulative distributions of the positive (a) and the negative (b) tails of the normalized returns of the Shanghai and Shenzhen Stock Exchange for an 8-year period from 3 January 1994 to 31 December 2001. The solid curve is a power–law fit in the region 3ogo10: We find a ¼ 2:44  0:02 for the positive tail, and a ¼ 4:29  0:04 for the negative tail.

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with a larger exponent than the negative tail does. To explore the detailed feature, one needs to accumulate more data for calculation. To confirm the relevant results, we also perform the Hill estimator method on our database. First, the normalized returns g are sorted in descending order, and these sorted returns are denoted as gk ; k ¼ 1; . . . ; N; where gk 4gkþ1 and N is the total number of events. Then the cumulative distribution is expressed in terms of the sorted returns as Pðg4gk Þ ¼ k=N and the inverse local slopes rðgÞ can be written as rðgk Þ ¼ 

lnðgkþ1 =gk Þ ’ k½lnðgK Þ  lnðgkþ1 Þ , ln½Pðgkþ1 Þ=Pðgk Þ

(5)

while k is large enough. Finally, the average value of all these data points yields the inverse slope r ¼ 1=a: Figs. 2(a) and (b) display these results. The average of the inverse slope gives the exponents with the value of ( 2:50 ðpositive tailÞ a’ . (6) 4:95 ðnegative tailÞ Such values match the previous results, but the fitting curves are not as perfect as in the power–law fit method, especially in the negative tail. This may be caused by the limitation of data points, since in the positive tail, which includes more data points, the fitting curve looks much more reasonable. For both tails the calculation region is set between 3ogo10; and each point represents an average over 100 inverse local slopes. We find that the positive tail contains almost 1.5 times data points as those of the negative tail. In addition, it should be noticed that in the positive tail, the data point near g ’ 10 gives a small value of r, which indicates that the value of a here will be larger than the above calculation results. This corresponds to the faster decay trend at the end of the tail in Fig. 1(a).

0.7

0.6

positive tail

r (Inverse exponent)

r (Inverse exponent)

0.6

0.7

0.5 0.4 0.3 0.2 0.1 0 0.1

(a)

negative tail

0.5 0.4 0.3 0.2 0.1

0.15

0.2

0.25

0.3

1/g (inverse normalized return)

0 0.1

0.35

(b)

0.15

0.2

0.25

0.3

0.35

1/g (inverse normalized return)

Fig. 2. Inverse slopes of the cumulative distributions of the normalized returns for Dt ¼ 1 day for the (a) positive and (b) negative tails. In either figure, each point is an average over 100 different inverse local slopes, and the solid line denotes the average value of all points for 3ogo10: The average of these points provides estimates for the tail exponent a ’ 2:50 (positive tail) and a ’ 4:99 (negative tail).

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Both estimates above for the asymptotic slope are in the region 3ogo12: For the region go3; regression fits yield smaller estimates of a; consistent with the possibility of a Le´vy distribution in the central region. To make sure this central behavior comparable with others, we also calculate the value of a in the region 0:5ogo3; and obtain ( a’

1:59 ðpositive tailÞ 1:82 ðnegative tailÞ

,

(7)

which are consistent with the result a ’ 1:4 found for small values of g in Ref. [6] and the result a ’ 1:6 in Ref. [13]. It indicates that in the center region, distribution behaviors are similar in both the Chinese stock market and those of the developed countries. To test the possibility of the exponential function behavior, we also try to fit the data points with the exponential function. But the fitting results are not as well as the power–law way. Therefore, we could conclude that the daily returns of Chinese stocks follows the power–law distribution. However, values of tail exponents are near 2.5 for the positive tail and 4.3 for the negative tail, which are quite different from the exponent around 3 for both tails discovered in stock markets of developed economics. To evaluate our conclusion, we investigate the histogram for a in Figs. 3(a) and (b). The data points are obtained from the individual cumulative distributions of all 104 companies studied in the region 1ogo10: The histogram gives median values of the exponent aM ’ 2:74 for the positive tail and aM ’ 4:05 for the negative tail. Thus, all these calculations match one another well. Based on all these analyses, the hypothesis that most stock markets follows the index of 3 rules does not match the Chinese stock market.

14

35 Number of occurrences

Number of occurrences

40

30 25 20 15 10 5 0

(a)

1

2

3 4 Exponent

5

12 10 8 6 4 2 0

6

(b)

2

2.5

3

3.5 4 Exponent

4.5

5

5.5

Fig. 3. The histograms of the power–law exponent of the positive (a) and the negative (b) tails obtained by the power–law regression fits to the individual cumulative distribution functions, where the fit is for 3ogo10: This histogram is not normalized, namely, the y-axis represents the number of occurrences of the exponent.

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Number of ocurrences

10

10

10

10

10

5

4

3

2

1

0

-15

-10

-5

0

5

10

15

g (normalized return)

Fig. 4. The distribution of g (14pgp14) vs. the number of occurrences, the distribution is narrower for small g and has a wider distribution for large g (solid line), compared with the Gaussian (dashed line).

Besides, we also perform a further study on the interesting discovery of the asymmetric tails. Fig. 4 depicts the distribution of g vs. the number of occurrences. It is apparent that in the distribution region 3ogo10; the negative tail descends faster than the corresponding positive tail. It demonstrates the data number difference between the two tails. As we refer to the definition of g, it is clear that a positive g reflects a rise on the price. According to the figure, it indicates that the Chinese stock market contains much more rise conditions, especially the large rises in prices in the area gX10: In addition, according to Fig. 4, the distribution is narrower for small g but wider for large g. This feature coincides with that of the Australian ‘All Ordinaries’ index [18]. Here, we do not discuss the results of varying time scales more than 1 day like in some other research work [13]. This is based on the fact that the data we have are so limited that expanding the time scale will bring out no more detailed results but only less data points for fitting.

3. Discussion It is well known that the Chinese stock market has only a history of a few more than 10 years: both Shanghai Stock Exchange and Shenzhen Stock Exchange were founded in 1990 and opened in 1991. This juvenility may be the reason why it behaves differently from those of highly developed countries. Here, we discover that the distribution of daily returns follows the power–law decay with tail exponents of around 2.5 for the positive tail and 4.3 for the negative tail, and exhibits an obvious asymmetric on its tails. These distinct features may be interpreted by the following reasons.

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First, the quantity of the data may affect the result. There is no theory that discusses the detailed relationship between the number of data and the final distribution behavior, but the quantitative difference of database between our calculation and other studies does exist. The study of the price distribution of the American stock market reaches the inverse-cubic law by the use of a data set around 107 [12], while in our studies, the quantity only reaches 2  105 : However, we check our results by the calculation of the individual stock exchange. In such a condition the number of data are around 2  104 ; but the results are almost the same, which indicates that under the quantity of 2  105 ; the change of the data number will not affect the distribution characteristic. Moreover, our data scale is the same as the study of the Indian stock market [22]. Second, such difference may be caused by the selected stocks themselves. Although we have selected all the stocks that came into stock markets in the early years, the existing time is still very short compared with most stocks in American and other developed economics. Whether or not the behavior of an individual stock will change as a function of its duration in the market still needs more studies. Finally, over speculation and fulsome impact of government policies may be the reason. As China is still a developing country, most economic institutions are still under construction and the stock market contains some immature facts. Meanwhile, small scale, instability as well as absence of shorting all contribute to the calculation results, especially the lack of short sale of Chinese stock markets results in the obvious asymmetry as shown in Figs. 1 and 4. Market behavior seems to be affected by these forces to perform such discontinuity and asymmetry. Two ways may be practised here: conform such infection as a removable error and try to take it off; or this is the real feature of Chinese stock market and may be a new law should be found. A further investigation based on a larger and more complete database is still under way.

Acknowledgements This project is partially supported by the State Key Development Programme for Basic Research of China (Grant No. 2001CB309308) and the Key Project of the Natural Science Foundation of the Ministry of Education of China (Grant No. 00-09).

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